A novel hybridity model for TiO2-CuO/water hybrid nanofluid flow over a static/moving wedge or corner

In this study, we are going to investigate semi-analytically the steady laminar incompressible two-dimensional boundary layer flow of a TiO2-CuO/water hybrid nanofluid over a static/moving wedge or corner that is called Falkner-Skan problem. A novel mass-based approach to one-phase hybrid nanofluid model that suggests both first and second nanoparticles as well as base fluid masses as the vital inputs to obtain the effective thermophysical properties of our hybrid nanofluid, has been presented. Other governing parameters are moving wedge/corner parameter (λ), Falkner-Skan power law parameter (m), shape factor parameter (n) and Prandtl number (Pr). The governing partial differential equations become dimensionless with help of similarity transformation method, so that we can solve them numerically using bvp4c built-in function by MATLAB. It is worthwhile to notice that, validation results exhibit an excellent agreement with already existing reports. Besides, it is shown that both hydrodynamic and thermal boundary layer thicknesses decrease with the second nanoparticle mass as well as Falkner-Skan power law parameter. Further, we understand our hybrid nanofluid has better thermal performance relative to its mono-nanofluid and base fluid, respectively. Moreover, a comparison between various values of nanoparticle shape factor and their effect on local heat transfer rate is presented. It is proven that the platelet shape of both particles (n1 = n2 = 5.7) leads to higher local Nusselt number in comparison with other shapes including sphere, brick and cylinder. Consequently, this algorithm can be applied to analyze the thermal performance of hybrid nanofluids in other different researches.


Problem Description and Governing Equations
Assume an incompressible laminar steady two-dimensional boundary layer flow over a static or moving wedge in an aqueous hybrid nanofluid with prescribed external flow and moving wedge velocities as displayed in Fig. 1. We have chosen titania (TiO 2 ) and copper oxide (CuO) as nanoparticles with water as base fluid. We also assume that the base fluid and nanoparticles are in thermal equilibrium and no slip occurs between them. It is worth mentioning that, to develop the targeted hybrid nanofluid TiO 2 -CuO/water, titania is initially dispersed into base fluid then, copper oxide is scattered in TiO 2 /water nanofluid. Therefore, the subscript (1) corresponds to first nanoparticle (TiO 2 ), while subscript (2) is applied for second nanoparticle (CuO) as well as subscript (f) related to base fluid. Table 1 shows thermophysical properties of the base fluid and the nanoparticles at 25 °C (see Dinarvand and Pop 45 , Nayak et al. 46 , Vajjha et al. 47 ).
According to Fig. 1, we choose 2D Cartesian coordinate system (x, y) where x and y are the coordinates measured along the surface of the wedge and normal to it, respectively. It is assumed that the free stream velocity is = ∞ U x U x ( ) m and the temperature of the ambient hybrid nanofluid is ∞ T , while the moving wedge velocity is = u x U x ( ) w w m and its constant temperature surface is . T w After using boundary layer approximations and Tiwari-Das nanofluid model (see Tiwari and Das 18 ) as well as the Bernoulli's equation in free stream, the governing non-linear PDEs of mass, momentum and energy can be written as follows (see Yacob et al. 42 ): w w In which u and v are the velocity components along x and y directions, respectively, T is the temperature of the hybrid nanofluid within the thermal boundary layer, C p is the specific heat at constant pressure, ρ hnf , μ hnf and   46 and Vajjha et al. 47 ). www.nature.com/scientificreports www.nature.com/scientificreports/ α hnf are the density, the viscosity and the thermal diffusivity of the hybrid nanofluid, respectively, and are defined according to Table 2.

Property Hybrid Nanofluid
In Table 2, k nf is the thermal conductivity of the single nanoparticle's nanofluid that is computed from Hamilton-Crosser model (see Ghadikolaei et al. 49 , and Hayat and Nadeem 50 where n is the empirical shape factor for the nanoparticle and is determined in Table 3, Moreover, we propose φ, ρ s and (C p ) s as the equivalent volume fraction for nanoparticles, the equivalent density of nanoparticles and the equivalent specific heat at constant pressure of nanoparticles, respectively, as well as φ 1 and φ 2 are solid fraction of first and second nanoparticles, respectively, that are calculated from following formulas (see Sundar et al. 48,52,53 ) P s P P 1 1 2 2  we notice that, w 1 , w 2 and w f are the first nanoparticle, the second nanoparticle and the base fluid masses, respectively. According to White 54 we are looking for a similarity solution of Eqs (1-3) along with boundary conditions (4) of the following form: where ψ is the dimensional stream function and is expressed in the usual form as ψ x / , f is the dimensionless stream function, θ is the dimensionless temperature distribution of the hybrid nanofluid and η is independent similarity variable. Fortunately, using similarity transformation method, substituting Eq. (11) into non-linear PDEs (2) and (3) and considering Eqs (8-10), give us a following set of dimensionless non-linear ODEs: Here, the Prandtl number (Pr), the constant moving wedge parameter λ ( ) as well as the Hartree pressure gradient parameter (β) are defined as w f f It should be mentioned that λ > 0 and λ < 0 correspond to a moving wedge in same and opposite directions to the free stream, respectively, while λ = 0 corresponds to a static wedge. Furthermore, β > 0 is caused by negative or favorable pressure gradient, while β < 0 creates positive or unfavorable pressure gradient (see White 54 ).
The skin friction coefficient C f and the local Nusselt number Nu x are defined as where, τ w is the shear stress at the surface of the wedge and q w is the heat flux from the surface of the wedge, which are illustrated by Finally, after combining Eqs (11), (17) and (18), we obtain Ux/ x f is the local Reynolds number. In summary, we can depict the computational procedure for our new algorithm in Fig. 2.

Results and Discussion
The similarity governing Eqs (12) and (13) along with boundary conditions (14) and (15) are solved numerically for some values of the governing parameters w 1 , w 2 , w f , φ, φ 1 , φ 2 , ρ s , (C p ) s , λ, m, n 1 , n 2 and Pr using the bvp4c built-in function from MATLAB software (see Shampine et al. 55 ). In this approach, we have considered η ≤ ≤ ∞ 4 6, η η ∆ = ∞ /100, and the relative tolerance was set as default (10 −3 ). Needless to say that, we are concentrating on the real (First) solutions that have correct physical reasons.
To validate our numerical procedure, Table 4 shows the value of the similarity skin friction coefficient (f ″(0)) for pure water φ φ φ = = = ( 0 ), 1 2 static boundary λ = ( 0) and different values of m. We can see from Table 4, with increasing parameter m the similarity skin friction coefficient enhances that it seems reasonable physically. Moreover, Table 5 42 and Nadeem et al. 44 . www.nature.com/scientificreports www.nature.com/scientificreports/ sponds to plane stagnation point flow case. It can be concluded that, with increasing m and w 2 both hydrodynamic and thermal boundary layer thicknesses decrease. So, the velocity as well as the temperature gradients enhance and according to Eqs (17) and (18)  and Pr = 6.2. As a result of this Figure, when the wedge moves in same direction to the free stream λ > ( 0), both hydrodynamic and thermal boundary layer thicknesses are thinner than the static wedge λ = ( 0) and moving wedge in opposite direction to the free stream λ < ( 0). So, the local Nusselt number is higher for moving wedges in same direction to the free stream λ > .

Hydrodynamic and thermal boundary layers.
( 0) Figure 2. Flowchart of the present problem's computational procedure. m Yih et al. 56 White 54 Ishak et al. 40 Yacob et al. 42 Nadeem et al. 44     and Pr = 6.2. It is quite clear that, both the skin friction coefficient (the undesirable effect) and the local Nusselt number (the desired effect) increase with increasing first and second nanoparticle masses for all cases. Indeed, increasing the nanoparticles mass leads to augmenting the effective thermal conductivity, and consequently tends the heat transfer rate enhancement of our heat transfer fluid. On the other hand, according to Eq. (19), the most important factors affecting the skin friction coefficient enhancement are (i) the first and second nanoparticles as well as the base fluid masses (w 1 , w 2 and w f ) and (ii) the absolute values of the dimensionless velocity profile's slope at the surface of the wedge (f ″(0)). As a result, the skin friction coefficient enhancement always can occur by net increase of both these factors. In HNF4 case, we obtain the largest heat transfer rate and also the maximum skin friction coefficient between all cases that means it has better heat transfer rate relative to single nanoparticle's nanofluid as well as pure water. So, the best status would be theoretically related to HNF4 case. Because in addition to having a 35% growth in heat transfer rate relative to pure water, it has a 36% increase in the skin friction coefficient. While HNF1, HNF2 and HNF3, respectively, have an increase in skin friction coefficient of about 17, 26, and 27%, compared to the base fluid. Nevertheless, checking the optimal range for these mass-based cases will require further field studies in the future. However, our major challenge is the high skin friction that requires the high pressure drop and the high relevant pumping power. Therefore, we always should control this issue for practical applications. After all, we can deduce that hybrid nanofluids sufficiently can be used in all applications where ever single nanoparticle's nanofluids have been used. Figure 7 demonstrates dimensionless temperature profiles for some values of nanoparticle's shape factor = n n ( ) 1 2 that were exhibited in Table 3   www.nature.com/scientificreports www.nature.com/scientificreports/ oparticles shape factor (n 1 and n 2 ) only affect the thermal characteristics of the problem due to their representations in the similarity energy Eq. (13) (see Eq. (13) and the thermal conductivity approximation of hybrid nanofluid in Table 2). On the other hand, values of local Nusselt number of nanoparticles shape factor from Fig. 7 are depicted in the bar diagram of Fig. 8. It is worth mentioning that, Fig. 8

Influence of nanoparticles shape on thermal characteristics of problem.
. Finally, in Table 6 we have compared the local Nusselt number of different shapes of first (TiO 2 ) and second (CuO) nanoparticles (n 1 and n 2 ) in terms of different hybrid nanofluid masses that are tabulated in Pr 6 2. It is seen that the local Nusselt number enhances with elevating shape factor of first or second nanoparticles in all cases. Further, it is perceived that, generally when the shape of second nanoparticle is spherical = n ( 3) 2 while, the shape of first nanoparticle is not spherical ≠ n ( 3), 1 the heat transfer rate of hybrid nanofluid is higher relative to opposite ones.

conclusions
The laminar two-dimensional Falkner-Skan problem by taking Newtonian TiO 2 -CuO/water hybrid nanofluid into account as the working liquid and with constant surface temperature was investigated semi-analytically with help of new proposed algorithm according to nanoparticles and base fluid masses. Our hypothesize was that the Prandtl number of water is 6.2. After implementing Tiwari-Das single-phase nanofluid model, non-dimensional form of the governing PDEs were written using auxiliary similarity variables, then we attempted to numerically solve them by bvp4c function from MATLAB. The major conclusions of this research, may be summarized as follows: (1) the Falkner-Skan power law parameter (m) and the second nanoparticle mass (w 2 ) increase the local Nusselt number at the surface of the wedge, (2) the local Nusselt number is higher for moving wedges in same direction to the free stream λ > ( 0) relative to static wedges λ = ( 0) as well as moving wedges in opposite direction to the free stream λ < ( 0), (3) mass increment of first and second nanoparticles invoke enhancement on skin friction and local heat transfer rate of our hybrid nanofluid, (4) when the nanoparticle shape is spheric, the local Nusselt number will be minimum than other nanoparticle shapes, (5) the HNF4 case with highest nanoparticles mass, possesses the largest local heat transfer rate between other mass-based cases, that means it has better thermal performance relative to mono-nanofluid and base fluid, respectively.     for some values of n 1 and n 2 based on various cases of hybrid nanofluids mass, when m 0 4, 02 λ = − . = . and = .